Adenosine triphosphate (ATP) utilization rate is directly related to the functionality of organ systems, such as contractility of the heart or neuron activity level in the brain. The study of ATP hydrolysis rate will help basic science researchers to investigate the energetic foundation of the function and dysfunction of certain organs. The ATP hydrolysis rate also has great potential in clinical applications. For instance, the ATP hydrolysis rate is tightly correlated to the severity of certain heart diseases (e.g. myocardial infarction), as shown by Q. Xiong, et al., in “Heterogeneity of ATP Turnover Rates in the LV of Swine Hearts with Post-Infarction Remodeling,” Circulation, 2012; 126:A12300. Thus, the ATP hydrolysis rate may serve as a quantitative metric to categorize heart disease patients for different treatment options, help characterize the severity of heart disease, predict the patient's prognosis, and indicate therapeutic efficacies of certain treatments.
Conventional magnetic resonance spectroscopy (MRS)-magnetization saturation transfer (MST) experiments have been used to measure ATP hydrolysis rate. The in vivo metabolism of ATP may be modeled as a chemical exchange network among phosphocreatine (PCr), ATP, and inorganic phosphate (Pi) (Equation 1):
                              PCr          ⁢                      ⇄                          k                              ATP                ->                PCr                                                    k                              PCr                ->                ATP                                              ⁢                      A            ⁢                                                  ⁢            T            ⁢                                                  ⁢            P                    ⁢                      ⇄                          k                              Pi                ->                ATP                                                    k                              ATP                ->                Pi                                              ⁢          Pi                ;                            [        1        ]            where the PCrATP exchange is catalyzed by creatine kinase (CK), the ATPPi exchange incorporates all other cellular activities that produce and consume ATP (Ugurbil, 2011), and k is the pseudo-first-order rate constant for the unidirectional reactions.
Conventionally, the evolution of visible MR magnetizations during chemical exchange are modeled with the modified Block-McConnell equations (Equations 2-4) (Bloch, 1946; McConnell, 1958):
                                                        d              ⁢                                                          ⁢                                                M                  PCr                                ⁡                                  (                  t                  )                                                                    d              ⁢                                                          ⁢              t                                =                                                                      M                                      0                    ,                    PCr                                                  -                                                      M                    PCr                                    ⁡                                      (                    t                    )                                                                              T                                  1                  ,                  PCr                                int                                      -                                          k                                  PCr                  ->                  ATP                                            ⁢                                                M                  PCr                                ⁡                                  (                  t                  )                                                      +                                          k                                  ATP                  ->                  PCr                                            ⁢                                                M                  ATPγ                                ⁡                                  (                  t                  )                                                                    ;                            [        2        ]                                                          ⁢                                                            d                ⁢                                                                  ⁢                                                      M                    Pi                                    ⁡                                      (                    t                    )                                                                              d                ⁢                                                                  ⁢                t                                      =                                                                                M                                          0                      ,                      Pi                                                        -                                                            M                      Pi                                        ⁡                                          (                      t                      )                                                                                        T                                      1                    ,                    Pi                                    int                                            -                                                k                                      Pi                    ->                    ATP                                                  ⁢                                                      M                    Pi                                    ⁡                                      (                    t                    )                                                              +                                                k                                      ATP                    ->                    Pi                                                  ⁢                                                      M                                          ATP                      ⁢                                                                                          ⁢                      γ                                                        ⁡                                      (                    t                    )                                                                                ;                                    [        3        ]                                                                    d              ⁢                                                          ⁢                                                M                                      ATP                    ⁢                                                                                  ⁢                    γ                                                  ⁡                                  (                  t                  )                                                                    d              ⁢                                                          ⁢              t                                =                                                                      M                                      0                    ,                                          ATP                      ⁢                                                                                          ⁢                      γ                                                                      -                                                      M                                          ATP                      ⁢                                                                                          ⁢                      γ                                                        ⁡                                      (                    t                    )                                                                              T                                  1                  ,                                      ATP                    ⁢                                                                                  ⁢                    γ                                                  int                                      +                                          k                                  PCr                  ->                  ATP                                            ⁢                                                M                  PCr                                ⁡                                  (                  t                  )                                                      +                                          k                                  Pi                  ->                  ATP                                            ⁢                                                M                  Pi                                ⁡                                  (                  t                  )                                                      -                                          (                                                      k                                          ATP                      ->                      PCr                                                        +                                      k                                          ATP                      ->                      Pi                                                                      )                            ⁢                                                M                                      ATP                    ⁢                                                                                  ⁢                    γ                                                  ⁡                                  (                  t                  )                                                                    ;                            [        4        ]            
By selectively saturating the terminal phosphate of ATP (ATPγ) magnetization during the MST experiment, Equations 2-4 change to Equations 5-7, respectively:
                                                        d              ⁢                                                          ⁢                                                M                  PCr                                ⁡                                  (                  t                  )                                                                    d              ⁢                                                          ⁢              t                                =                                                                      M                                      0                    ,                    PCr                                                  -                                                      M                    PCr                                    ⁡                                      (                    t                    )                                                                              T                                  1                  ,                  PCr                                int                                      -                                          k                                  PCr                  ->                  ATP                                            ⁢                                                M                  PCr                                ⁡                                  (                  t                  )                                                                    ;                            [        5        ]                                                                    d              ⁢                                                          ⁢                                                M                  Pi                                ⁡                                  (                  t                  )                                                                    d              ⁢                                                          ⁢              t                                =                                                                      M                                      0                    ,                    Pi                                                  -                                                      M                    Pi                                    ⁡                                      (                    t                    )                                                                              T                                  1                  ,                  Pi                                int                                      -                                          k                                  Pi                  ->                  ATP                                            ⁢                                                M                  Pi                                ⁡                                  (                  t                  )                                                                    ;                            [        6        ]                                                      M                          ATP              ⁢                                                          ⁢              γ                                ⁡                      (            t            )                          =        0.                            [        7        ]            
Equations 5 and 6 are mathematically equivalent, describing the unidirectional kinetics of ATP production from PCr and Pi, respectively. Solving Equations 5 and 6 yields Equation 8:
                                          k                                          PCr                ⁡                                  (                  Pi                  )                                            ->              ATP                                =                                    (                                                                    M                                          0                      ,                                              PCr                        ⁡                                                  (                          Pi                          )                                                                                                      -                                      M                                          ss                      ,                                              PCr                        ⁡                                                  (                          Pi                          )                                                                                                                                      M                                      ss                    ,                                          PCr                      ⁡                                              (                        Pi                        )                                                                                                        )                        /                          T                              1                ,                                  PCr                  ⁡                                      (                    Pi                    )                                                              int                                      ;                            [        8        ]            where Mss and M0 represent the fully relaxed magnetizations with and without saturation of ATPγ, and T1int is the intrinsic longitudinal relaxation time constant. Thus, the calculation of the ATP production rate constants via Equation 8 requires two fully relaxed spectra: one control spectrum without saturation to obtain M0,PCr(Pi) and one saturated spectrum with the saturation pulse set at the ATPγ frequency to obtain Mss,PCr(Pi). Intrinsic T1 (T1int) is usually a constant among patients and T1int is unaffected by physiological or pathological conditions; however, if T1int is not known, it may be calculated from another mathematically equivalent Equation 9:
                                          k                                          PCr                ⁡                                  (                  Pi                  )                                            ->              ATP                                =                                    (                                                                    M                                          0                      ,                                              PCr                        ⁡                                                  (                          Pi                          )                                                                                                      -                                      M                                          ss                      ,                                              PCr                        ⁡                                                  (                          Pi                          )                                                                                                                                      M                                      0                    ,                                          PCr                      ⁡                                              (                        Pi                        )                                                                                                        )                        /                          T                              1                ,                                  PCr                  ⁡                                      (                    Pi                    )                                                              app                                      ;                            [        9        ]            where T1app is measured directly via progressive saturation or inversion recovery acquisitions and T1app is then used to calculate T1int according to the following Equation 10:(T1app)−1=(T1int)−1+k  [10].
As shown in Equation 9, the conventional MST approach for measuring the in vivo ATPPi rate constant relies on the quantification of Pi magnetization levels (MPi). It remains difficult to quantify in vivo Pi magnetization levels (MPi), especially for certain organ systems such as in vivo heart, because myocardial Pi levels are intrinsically low and the peaks of Pi and 2,3-diphosphoglycerate (2,3-DPG), which is generated from erythrocytes in the blood, at least partially overlap. Furthermore, there is a lack of consensus about whether intracellular cardiac Pi is completely visible in an MR spectrum (Jeffrey, Storey, 1989; Humphrey and Garlick, 1991).
Thus, there remains a need for a method of measuring the unidirectional ATP→Pi rate constant using magnetic resonance without requiring the direct, quantification of inorganic phosphate (Pi).